What are examples of simple problems and applications that can be solved with AI techniques?
This is fairly boilerplate advice, but, since you're brand new to AI, I'd personally suggest writing a classical Tic-Tac-Toe AI, ideally using minimax.

I suggest this because minimax is fundamental to AI, and there are many webpages devoted to this subject, such as How to make your Tic Tac Toe game unbeatable by using the minimax algorithm and Tic Tac Toe: Understanding the Minimax Algorithm.  (Google search for "Tic-tac-toe" and "minimax" will yield a plethora of other sites.  I'd also recommend looking at this minimax page from Stanford: "Strategies and Tactics for Intelligent Search".)

I recommend this approach as a good basic primer.  The real cutting-edge work is being done in Machine Learning and Neural Networks, and for that reason, it's probably more important than ever to have some basic grounding in classical AI before you start dipping your toe in that pond.